Question: The lifespans of gorillas in a particular zoo are normally distributed. The average gorilla lives $20.8$ years; the standard deviation is $3.1$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a gorilla living between $11.5$ and $27$ years.
Explanation: $20.8$ $17.7$ $23.9$ $14.6$ $27$ $11.5$ $30.1$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $20.8$ years. We know the standard deviation is $3.1$ years, so one standard deviation below the mean is $17.7$ years and one standard deviation above the mean is $23.9$ years. Two standard deviations below the mean is $14.6$ years and two standard deviations above the mean is $27$ years. Three standard deviations below the mean is $11.5$ years and three standard deviations above the mean is $30.1$ years. We are interested in the probability of a gorilla living between $11.5$ and $27$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the gorillas will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the gorillas will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of gorillas between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular gorilla living between $11.5$ and $27$ years is $\color{orange}{2.35\%} + {95\%}$, or $97.35\%$.